CFA二级-复习冲刺-数量.docx

Contents:StudySession3:QuantitativeMethodsReading6:FintechInvestmentManagement()Reading7:CorrelationandRegression()Reading8:MultipleRegressionandIssuesinRegressionAnalysis()Reading9:Time-SeriesAnalysis()Reading10:Excerptfrom"'ProbabilisticApproaches:ScenarioAnalysis,DecisionTrees,andSimulations"()u-EeIzoraraOM-06tj3duol-<三-ol-traspOiAl>一ss-g-号V:->-euvipitedSPOIIl-H-<S-J'E二-S七3S-SA-BUXZUO-SSBJIgaHU-Sanss-ZbtedS-S>eu<4>0N4-mteduo55Hc-d-PUe-dlu一SNt3dS-SA-BUXZcora-<-ouSPOWaIAj>racra2Q:mUo-SsasAPnlS:匕UBwoUSeSUSVefiS-SsCorreIationAnaIysisCorrelationanalysisScatterplotsSamplecovarianceandcorrelationcoefficientCov(X,Y)Xi-XYi-YCov(XJ)=Fn-lHypothesistestingofcorrelation1.imitationstocorrelationanalysisOutlierSpuriouscorrelationNonlinearrelationshipsCorrelationAnalysisHypothesistestingofcorrelationTestthecorrelationcoefficientbetweentwovariablesisequaltozero.Hp=0,H:aPH.t-test:t=ri二2df=n-2.Two-tailedtest.RejectHf:t>+tcritical,。t<-tcriticalp-value<significancelevel().707工SNOOO9I工、ON(89z工&(4SE°JlU左PA-IUeaJ-UNSu三ooIlo-I-Uo-ucrau三cO&>-SlvZSOMOIU£OuUo一一0。B>£S0usUo4JBs-rol-roE-urooororoqoSPUq-EOJoAPn<Practice1Answer:AH0:r=Ovs.Ha:rO;tri0.52312l-O.522t(=0.05anddf=38)=2.021Ct>thencewerejectH0.IUp三oUOSS8匿BOJIS2ucrao三Slupso3Uo-SSBg3JoUO4el-e-lu-SUo-IBnbpuroS-PoEUo-SSjJClJOmUo-SS(Sio)S3nbsISB-AJeU-PJOSUo4dlunss<Uo万sajtwo-e=-d=3EPUe-dESe-MMSSK3una三s三1«3lu-sSimpleandMultipleLinearRegressionAssumptionsTherelationshipbetweenthedependentvariableandtheindependentvariablesislinear.Theindependentvariablesarenotrandom.Formultiplelinearregression,noexactlinearrelationexistsbetweentwoormoreoftheindependentvariables.impleandMultipleLinearRegressionAssumptions(Cont.)Theerrortermisuncorrelatedacrossobservations:E(JEj)=Oforallij.Theerrortermisnormallydistributed.Theexpectedvalueoftheerrorterm,conditionedontheindependentvariables,is0:E(XpX2,.,Xk)=0.Thevarianceoftheerrortermisthesameforallobservations(homoscedasticity,同方差性):E()=.-UOSSPUeSUo43JesqoWUNOMWqra-B-t->PSBrlbSOUJnS:LIJ(IJJSOBPsenbsjoEnS一ON-EZ-E0一Bdo-SPUBIdEllJoJ>oou三O-SStwJSlouss-(S"lo)Salenbslsea-A-Bu-P-Osos三a三lv三三s三IraRegressionmodelsandequationsForsimplelinearregression:Yj=b0+b1Xi+jV=b°+bYFormultiplelinearregression:Y=b+bXj+bX)+.+bkXki+JIUI11Cqlr=b0b1X1i+b2X2+bkXkiSimpleandMultipleLinearRegressionInterpretationofregressioncoefficients/VSlopecoefficient(b)j:sensitivityofYtoachangeinX.Forsimpleregression:changeofYfor1-unitchangeofX.Formultipleregression:changeofYfor1-unitchangeof%,holdingtheotherindependentvariablesconstant.Interceptcoefficient(b*Forsimpleregression:valueofYwhenXiszero.Formultipleregression:valueofYwhentheindependentvariablesareallzero.SimpleandMultipleLinearRegressionSignificancetestforaregressioncoefficientHb0；H:a*0bTeststatistic:t=1df=-k-l.s-RejectHf:bt>+tcritical/Ort<-tcriticalp-value<significancelevel().RejectionofH(f11eansbjssignificantlydifferentfromO.XfxplainssomevariationofY.-ras-8u(PgUoU-lBp-号。-On-e>NBO-lH一员jx-一一q上(gsX。一)qjqv(<sxJ),.-qjo(<SXJ)÷lq<V<qJoJ-ro-ooOql(luou)IUM=JaouUO-SSaJMa-BOJIS一。UeJM=EWSUO-SSaIbaII三=0三三三三1Practice1ConsidertheregressionresultsfromtheregressionofYagainstXfor50observations:Y=0.78+1.2X.Thestandarderroroftheestimateis0.40andthestandarderrorofthecoefficientis0.45.Whichofthefollowingreportsthecorrectvalueofthet-statisticfortheslopeandcorrectlyevaluatesitsstatisticalsignificancewith95%confidence?A)t=3.000;slopeissignificantlydifferentfromzero.B)t=2.667;slopeissignificantlydifferentfromzeroC)t=1.789;slopeisnotsignificantlydifferentfromzero.0ZEoIUaJNJ-P-do-sl(Joj8Jllloz虫BuJoP(UJOl-8寸OJffi->trs三LK99ZUSt7。/(0ZU)HlS-1S4-SISSql°ZE2“=之-P*IU-七u0uBdo-SajLl-OU一Liulp0一2roUJJo七dCQCMSUV1.33=3三dU一一S+:口sllZHw<PUBZaGLLIS)leE-S3O2°P-BpUBlSqsOONV>-ro<>oz<S-SAIwloNVANOVAtabledfSumOfSquares(SS)MeanSumofSquares(MS)Regression(explained)kRSSMSR=RSSkError(unexplained)-klSSEMSE=SSE(n-k-l)Total11-1SST=RSS+SSE-U一一0tqJB=BESU=Uo-SSbj0aJlJo-一sebj0n3DULUSU0-SSCUQIlIU-SuJJl0JoUo-I3>pPJ3purolsHUJLUS)“0<-otPjiBPUBASS-SAIWUINUR2andAdjustedR2Rz(coefficientOfdetermination):thepercentageofthetotalvariationthatisexplainedbytheregression.r2_Explainedvariation_RSS_SST-SSETotalvariationSSTSSTTesttheoveralleffectivenessoftheregressionmodel.Thehigher,thebetter.However,R2willincreasesimplybyaddingindependentvariables,eveniftheaddedoneisnotstatisticallysignificant.R2andAdjustedR2(C0nt.)AddinganewindependentvariablemayeitherincreaseordecreasetheadjustedR2.2-(nT)1R2AdjustedR1-In-k-"-_AdjustedR2R2,andmaylessthanzeroifR2islowenough.WhenaddinganewvariablewithsmalleffectonR2,thevalueofadjustedR2maydecrease,andthenewvariableshouldnotbeadded.F-statisticsAssesstheeffectivenessoftheindependentvariablesasagroup,orthemodelasawhole,inexplainingthedependentvariable.Testwhetherallregressioncoefficientsaresimultaneouslyequaltozero.HlbqbIb与.=b吊0H：aatleastoneb0(j=1tok)1.MSRMSEF-statistics(Cont.)Decisionrule:rejectH秣F-statistic>F-criticalvalue.TheF-testhereisalwaysaone-tailedtest.RejectionofHt511eansthereisatleastoneregressioncoefficientissignificantlydifferentfromzero.Atleastoneindependentvariablemakesasignificantcontributiontotheexplanationofdependentvariable.Forsimplelinearregression,theF-testduplicatethet-testforthesignificanceoftheslopcoefficient.H:ObKO,H:bq0.(冰要皿(SS-H整)UOUPBdSSfOJ-qBzEAIUBPUjdBPA-IE±-3nQ-qe一e>Ulapuadapu-EW-enor(史舞#«/)AWsw=Ou一n,冰要IEb-0ro-l-l-005<UO=-三Xlou-eMas(举溜K)AlP=SePgKMgBHSUOAdEnSSeOSUo-ABo>S-SA-BUBUOSSJMaJ.三Sanss-S-SA-BlIUIiCHSSaIES-S3nss_Heteroskedasticity:definitionVarianceoftheerrorsdiffersacrossobservations.UnconditionalHeteroskedasticity:theerrorvarianceisnotcorrelatedwiththeindependentvariables.Createsnomajorproblemsforstatisticalinference.ConditionalKeteroskedasticity:theerrorvarianceiscorrelatedwiththevaluesoftheindependentvariables.Doescreatesignificantproblemsforstatisticalinference.Heteroskedasticity:effectsThecoefficientestimates(b)jaren'taffected.Thestandarderrorsofcoefficient(Sqareusuallyunreliable.Withfinancialdata,thestandarderrorsaremostb-blikelyunderestimate,andthet-statistics(t-)Sawillbeinflated,suggestingsignificancewhenthereisnone(typeIerror).TheF-testisalsounreliable.poluUo一SSbolp=nqOIS3nbsIS3-pz=JUOJOs->SJOJJPJBPUCnqOJM.三ti±oo>lo4sep>lso-lHAlpAsepBXSO-BlBlI0H4S9ZXoran-ws-ranp一SJ-oSiO-dl-rotwUASalXlP=SePeXSO-BleHS-SA-EIIVU*三SS3三3H三SalISS-WEOJUBJMlJqu04iz03.arocjj0U-IJo-Iroi=03-roS>2El-0e5-cora-too-sodIUORe-BXIOY-roS仁.-sod.Suo-SsSs山E4U-MadAlPUeU一ouraUoI-Ms-l-oo3SILU一0IIlUoA-fcsp-UORe-BXIOJ-B=BSS-SA-EIIV!.三ss0s三SalISS-qe-JUnOS-BS-sl-l-Hdo。-BdAl)IUes三u°psou-zro-l-ra>rolPnpUoO-1。0。U-'poUJ4sJPUnS-1S4B1SJPUBPele-JU-SJOJJepropuels-UOQB-B-UOo-3s>s(0-BdAl)UoU-°一UCMootMwEelJU一s-sspu3Ee-uSOBro_uro三o-l-l-ou-Js>0qe=UnA=BnSn°BIU-JO3Os0ZePUBlSBlllpo:P=rasSelBIJU-ISBlu:BoS3111SIo4=:uo-rooo。-e-,lsS-SA-BIIUU*三SS3三3H三Sallss-Serialcorrelation:testingScatterplotsoftheresiduals.TheDurbin-Watson(DW)test.HNoserialcorrelationDecisionrule:In-FailtoIn-PositiveconclusiverejectH0conclusiveNegativeDW=OdLd4-d4-clDW=4(r=l)(r=-l).JS1-Uo4Bnbot-l-BiiiAJ-POIAIs0PJ5puelsIUBP-JJBo0一lsnfp<cu-ouEORe-BJUOJ-£BSSAUV!.三ss0s三SalISS-(0OJ-dAl)lu3u三uyslouS-qraB>PIlPnpUoO-:P0。U-、p：IeEASPUnSalSAelSJPllBpelSJO-uPjlePUSSqe=8unPUBsp-UJ一-UJJ×EOOO-SlU-JJOjUo-SS°CIjOJJoSEc=ou三3-P3e£Oj-三Bs-qBe>IUBPUBdBPU-UO三UF3Pc=-0u三3IS-SA-EIIVU*三SS3三3H三SalISS-s-qe>lupudpu-pl-uo°lJOEOE00M-O×M.三c0。：AweU=-O-HnlAloura三UaS-BJ>orau-VM-PUBIl-llS-ZH-WM、(一SIJ)IUeJ三u-s-SIU-JOJUo-SS七OUoNMun:-一Jle'e=84sS-SA-EIIVU*三SS3三3H三SalISS-SummaryofassumptionviolationsViolationEffectsTestingConditionalHeteroskedasticity与»TypeIerrorBreusch-Pagen2-testH0:noHeteroskedasticityPositiveserialcorrelationsj，TypeIerrorDurbin-WatsontestH0:noserialcorrelation0DWcIl:positivecorrelationNegativeserialcorrelations/TypeIlerrorMulticollinearitys/TypeIlerrorNonet-testssignificant,F-testsignificantandhighR2S-qraVB>AEUJnPTlJPUMosoOEra-t-3OllUBMM、/L0JO->EUoSBaElPUE-qrae>>ra-JoBd-Uo-qBe>AEUJna-qe-Jle>UIaPUadaPU->ra-ra3oS-SA-EIIVU*三SS3三3H三SalISS-Qualitativeindependentvariable(Cont.)Example:Ybb,#b,2f-b妙Eiwhere:Yj=quarterlyvalueofEPSofastock.YXiX2X3QIEPS100Q2EPS010Q3EPS001Q4EPS(omittedcategory)000Qualitativeindependentvariable(Cont.)Interpretationofcoefficient:Interceptcoefficient:theaveragevalueofdependentvariablefortheomittedcategory.Regressioncoefficient:thedifferenceindependentvariable(onaverage)betweenthecategoryrepresentedbythedummyvariableandtheomittedcategory.Practice1WhichofthefollowingisleastaccurateregardingtheDurbin-Watson(DW)teststatistic?A)Iftheresidualshavenegativeserialcorrelation,theDWstatisticwillbegreaterthan2.B)IntestsofserialcorrelationusingtheDWstatistic,thereisarejectionregion,aregionoverwhichthetestcanfailtorejectthenull,andaninconclusiveregion.C)Iftheresidualshavepositiveserialcorrelation,theDWstatisticwillbegreaterthan2.>snpuoCJu-S-HJllpzMU-n-B>Jowl-ra-0Illuo4113wo3>-o-BsZro-l-SS-ro>Bpu3-ora-t-<-oo>s-3-PU-ZUB-l-rol-M-ro>ro-oro-l-l-oOUroo-ZJo-ra><U:MSUVModelTrendmodels1.ineartrendmodels:constantchangeamount101y=b+btLog-lineartrendmodels:constantgrowthrateIn yt =b0+ b1t + t_a(b°+bt+Et)1.imitationsoftrendmodelsNotappropriatewhenserialcorrelationexists.AssumptionCovariancestationaryisakeyassumptionforARtimeseriesmodeltobevalidbasedonOLSestimates.Acovariancestationaryseriesmustsatisfy:Constantandfiniteexpectedvalueinallperiods.Constantandfinitevarianceinallperiods.Constantandfinitecovariancewithitselfforafixednumberofperiodsinthepastorfutureinallperiods.-AutoregressivemodelAutoregressivemodelxt=b0÷blxt.1+txt三b0+b1xl+b2xt.2+.+bpxt.p+lChainruleofforecastingt÷=+b1xlxt+2=b0+b1xt+1AutocorrelationAutocorrelationdetectiont-testofautocorrelations(pIk)tSeasonality:showsregularpatternsofmovementwithintheyear.Testing:t-testofseasonalautocorrelation.The4thautocorrelationincaseofquarterlydata.The12thautocorrelationincaseofmonthlydata.Correcting:includeaseasonallag.-POUJ(I)X<UBJOJ>-t><->ro-roESI-spBMole>OUJ一Uo-S>-Uo2>jlUBBw-三azssalOMoPoIEUo-SSSBnbSIS3-HAJeUO-IelSoro>ooo-z×IoU=-Ma-BMEoPUe<(Io0I-UnIl-M(I)cx-<)PoUJ>l-3MUJoPUBXX-EEoPUBH_暑0Hl02vu三Dickey-FullertestforunitrootStep1:startwithanAR(I)model:Xqbbt.tStep2:subtractxt.frombothsides:X辛ib(brl)xt.tOr:X辛t.fbg占ttwhere:g1=b1-lStep3:testifg=j.Hbg;H:ag割Calculatet-statisticanduserevisedcriticalvalues.IffailtorejectHthereisaunitrootandthetimeseriesisnon-stationary.FirstdifferencingArandomwalk(i.e.,hasaunitroot)canbetransformedtoacovariancestationarytimeseriesbyfirstdifferencing.SubtractXt.frombothsidesofrandomwalkmodel:×t-×M=×t-×M+三t=三tDefineyqx辛心。yqendbecomecovariancestationary.QBJrmeISoUJ-0MadAl3l3pv-dEesc?Inol-0-IS=BESlllLI-MPOE一wosjSsP3enbsueoEIooHPOUJ一raEOIPoyd-O-Ep-slnoS-BnP-S一50S3seu3-OJQdUJeSCOJnoPOUJ(IJE0一pod-O-EraUN-M-BnP-SBJBlIISJOtBSlSBJa-O-QHEra?c-Uo=Bn-e>POII-三azssalReliabilityvs.stabilityTheestimatesofregressioncoefficientsofthetime-seriesmodelcanchangeacrossdifferentsampleperiods.Theisatradeoffbetweenreliabilityandstability.Modelsestimatedwithshortertimeseriesareusuallymorestablebutlessreliable.psnNqIoUraoUo-SSajJBU=、psuo3：IoUJn3SgJBSOMl一-psnaqCsUO-SS二BU=Eg3：MJO3n3SSOMI一-IooJIn>rosSE0qpcsno-IoUCraoUo-SSOJJelJOaJJBu-qoo-UnroSeils-JsE0-Uopsn-sQqUBUUO-SSeJga-JeU=Ioo-UnSBllsSE一OUoUJ-s-,lsEnOMlq-MUo-SSalmaH-ttl三BwsSaI=WlIIVBarryPhillips,CFA,isanalyzingquarterlydata.HehasestimatedanAR(I)relationship(×t=b0+b1×xt.1+et)andwantstotestforseasonality.Todothishewouldwanttoseeifwhichofthefollowingstatisticsissignificantlydifferentfromzero?A)Correlationfeveb4)B)Correlation(et,et.1).C)Correlation(et,et.5).Answer:AAlthoughseasonalitycanmaketheothercorrelationssignificant,thefocusshouldbeoncorrelation(eet,4)becausethe4thlagisthevaluethatcorrespondstothesameseasonasthepredictedvariableintheanalysisOfquarterlydata.VsdH1.OoloIHOqITqra-rorolou8¾PISIoUuIVLnHoqSPIB°q3leplJU°JJ-PIS上JIoU(8IH,-lqSnBjqBlBP°U8JEpIS上J(VE三。-ssd=三d+×0+0HI+×u-nsJ->0=0Spu3dzsuow-aj<uroPSeUJASBsroll<:OASd=IIdt-<-Z33=3三d